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Chapter 7: Problem 58

A tandem (two-person) bicycle team must overcome a force of \(165 \mathrm{~N}\) to maintain a speed of \(9.00 \mathrm{~m} / \mathrm{s}\). Find the power required per rider, assuming that each contributes equally.

Short Answer

Expert verified
The power required per rider is 742.5 W.

Step by step solution

01

Understanding Power

Power is defined as the rate at which work is done. The formula for power is given by the expression \( P = F \times v \), where \( P \) is the power, \( F \) is the force, and \( v \) is the velocity.
02

Calculate Total Power Required

To find the total power required to overcome the force, we use the formula \( P = F \times v \), where \( F = 165 \, \mathrm{N} \) and \( v = 9.00 \, \mathrm{m/s} \). Thus, the total power needed is \( P = 165 \, \mathrm{N} \times 9.00 \, \mathrm{m/s} = 1485 \, \mathrm{W} \).
03

Determine Power Per Rider

Since the question states that each rider contributes equally, we divide the total power by the number of riders. There are two riders, so the power per rider is \( \frac{1485 \, \mathrm{W}}{2} = 742.5 \, \mathrm{W} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Calculation
Power refers to how quickly work is done, or energy is transferred. It's a measure of efficiency concerning time. In physics, we use the formula for power as:
  • Power (\( P \)) = Force (\( F \)) 脳 Velocity (\( v \)).
This formula clearly shows that power is a combination of both force and motion. Here, force is applied to maintain a certain velocity, which implies consistent work is being done.
When you apply this formula in a problem like a tandem bicycle team, it measures the total energy output needed by the bicyclists to maintain speed against given resistance. This is useful in determining how much effort each rider needs to exert under specific conditions.
Force and Motion
Force and motion are fundamental concepts in physics governing why objects move or stay still. They always go hand in hand. Force is any interaction that can change the motion of an object.
In this exercise, the force \( 165 \mathrm{~N} \) is what the cyclists need to overcome to keep their bike moving at a steady rate.
  • Force acts in the direction of the motion.
  • Maintaining constant velocity means that the force applied by the bicyclists exactly balances the opposing forces, like friction.
Understanding this balance and how forces work together is crucial in determining the efforts needed for any motion.
Work and Energy
Work and energy cover how forces cause motion. Work is done when a force causes movement, while energy is the capacity to do work.
  • Work (\( W \)) is calculated by multiplying the force applied by the distance over which it acts.
In our scenario, while the solution focuses on the power and velocity aspects, work and energy are foundational. The power calculation ultimately derives from an understanding of how much work is being done per unit of time.
It connects the dots between force exerted, movement through cycling, and the energy required to sustain that motion. Thus, grasping work and energy helps you see the bigger picture of why achieving constant velocity requires continuous exertion.
Velocity
Velocity is speed with direction. It describes how fast an object is moving and in which direction. Unlike speed, velocity is a vector.
  • In the discussed tandem bicycle problem, the velocity is constant at \( 9.00 \mathrm{~m/s} \).
  • This consistency indicates a steady state where input and opposing forces are matched.
Understanding velocity is key to solving power-related problems because the relationship between velocity and force directly influences how much power will be needed. It reflects the energy needed to maintain the current state of motion despite opposing forces, like air resistance or friction. Recognizing how velocity interplays with force can shed light on the practicalities of real-world motion scenarios.

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Most popular questions from this chapter

A block with mass \(0.50 \mathrm{~kg}\) is forced against a horizontal spring of negligible mass, compressing the spring a distance of \(0.20 \mathrm{~m}\), as shown in Figure \(7.47 .\) When released, the block moves on a horizontal tabletop for \(1.00 \mathrm{~m}\) before coming to rest. The spring constant \(k\) is \(100 \mathrm{~N} / \mathrm{m} .\) What is the coefficient of kinetic friction \(\mu_{\mathrm{k}}\) between the block and the tabletop?

The engine of a motorboat delivers \(30.0 \mathrm{~kW}\) to the propeller while the boat is moving at \(15.0 \mathrm{~m} / \mathrm{s}\). What would be the tension in the towline if the boat were being towed at the same speed?

A good workout. You overindulged in a delicious dessert, so you plan to work off the extra calories at the gym. To accomplish this, you decide to do a series of arm raises while holding a \(5.0 \mathrm{~kg}\) weight in one hand. The distance from your elbow to the weight is \(35 \mathrm{~cm}\), and in each arm raise you start with your arm horizontal and pivot it until it is vertical. Assume that the weight of your arm is small enough compared with the weight you are lifting that you can ignore it. As is typical, your muscles are \(20 \%\) efficient in converting the food energy into mechanical energy, with the rest going into heat. If your dessert contained 350 food calories, how many arm raises must you do to work off these calories? Is it realistic to do them all in one session?

U.S. power use. The total consumption of electrical energy in the United States is about \(1.0 \times 10^{19}\) joules per year. (a) Express this rate in watts and kilowatts. (b) If the U.S. population is about 320 million people, what is the average rate of electrical energy consumption per person?

A boat with a horizontal tow rope pulls a water skier. She skis off to the side, so the rope makes an angle of \(15.0^{\circ}\) with the forward direction of motion. If the tension in the rope is \(180 \mathrm{~N},\) how much work does the rope do on the skier during a forward displacement of \(300.0 \mathrm{~m} ?\)
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